I'd actually have you not push the hips back first - it tends to overemphasize a backward hip drive, and that's the last thing you want in an OHS or FS. I think your bottom position woes are most likely a product of your head position and the fact that your pushing your hips back.
I can't see where your eyes are directed, but based on your head position, it looks like you're looking down - don't. Look straight ahead. Keep the head push forward through your arms like you're doing, but keep it a little more upright, and not quite as far forward. Driving it down and too far forward will invariably pull your chest down like that.
I think your foot width is fine - you're getting plenty of external rotation of the hips and space to sink down there - if you go much wider, your knees are going to be twisting. But in that bottom position, you need to really sit your hips down onto your heels as much as possible - bring the torso upright just short of vertical (to allow the correct shoulder positioning). Again, I think this is tied to your head pulling you forward.